Question

pls answer this question and I will mark you as brainliest​

Answer 1

Factor Theorem:-

A polynomial p(x) is exactly divisible by (x - a), or has a factor (x - a), if and only if p(a) = 0.

Proof:-

Let a polynomial p(x) be in such a way that,

→   p(x) = (x - a) q(x)

It means p(x) divided by (x - a) gives the quotient q(x) but no remainder. That is, p(x) is considered to be exactly divisible by (x - a) for some real number a.

Here x is a variable so that it can accept every real number values. Hence x = a is possible too, since a is a real number.

If  x = a,  we see that,

→   p(a) = (a - a) q(a)

→   p(a) = 0 × q(a)

→   p(a) = 0

This means the condition that p(a) = 0 should be satisfied for the term (x - a) strictly being a factor of p(x).

Hence the Proof!

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Here,

→   p(n) = n³ - 8

We have to show that (n - 2) is a factor of (n³ - 8).

For (n - 2) being a factor, p(2) should be zero.

→   p(2) = 2³ - 8

→   p(2) = 8 - 8

→   p(2) = 0

So p(2) is equal to zero. Hence it is true that (n - 2) is a factor of (n³ - 8).